Question

If `x^4 + 1/x^4 = 194, "find"  x^3 + 1/x^3`

Answer 1

In the given problem, we have to find the value of  `x^3 + 1/x^3.`

Given: `x^4 + 1/x^4 = 194`

We know that,

`(x^2 + 1/x^2)^2 = x^4 + 1/x^4 + 2`   ...[Using (a + b)² = a² + b² + 2ab]

Now, substituting the given value

`(x^2 + 1/x^2)^2 = 194 + 2`

∴ `(x^2 + 1/x^2)^2 = 196`

∴ `x^2 + 1/x^2 = sqrt196`

∴ `x^2 + 1/x^2 = +-14`

Let’s relate it to `x^3 + 1/x^3`

`(x + 1/x)^2 = x^2 + 1/x^2 + 2`   ...[Using (a + b)² = a² + b² + 2ab]

`(x + 1/x)^2 = 14 + 2`

∴ `x + 1/x = sqrt16`

∴ `x + 1/x = +-sqrt4`

Thus, 

`x^3 + 1/x^3 = (x + 1/x)^3 - 3(x + 1/x)`

`x^3 + 1/x^3 = (4)^3 - 3(4)`

`x^3 + 1/x^3 = 64 - 12`

∴ `x^3 + 1/x^3 = +-52`